Research

Nontrivial solutions for a boundary value problem with integral boundary conditions

Bingmei Liu1*, Junling Li1 and Lishan Liu2

Author Affiliations

1 College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China

2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China

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Boundary Value Problems 2014, 2014:15  doi:10.1186/1687-2770-2014-15

 Received: 27 July 2013 Accepted: 13 November 2013 Published: 13 January 2014

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a sign-changing continuous function and may be unbounded from below.

1 Introduction

Consider the following Sturm-Liouville problem with integral boundary conditions

(1.1)

where , , , , , α and β are right continuous on , left continuous at and nondecreasing on with ; , and denote the Riemann-Stieltjes integral of u with respect to α and β, respectively. Here the nonlinear term is a continuous sign-changing function and f may be unbounded from below, with is continuous and is allowed to be singular at .

Problems with integral boundary conditions arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3]. Integral BCs (BCs denotes boundary conditions) cover multi-point BCs and nonlocal BCs as special cases and have attracted great attention, see [4-14] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [4], Agarwal and O’Regan [5]. Yang [6], Boucherif [8], Chamberlain et al.[10], Feng [11], Jiang et al.[14] focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated two-point and multi-point boundary value problems with sign-changing nonlinear terms, for example, [15-20], results for boundary value problems with integral boundary conditions when the nonlinear term is sign-changing are rarely seen except for a few special cases [7,12,13].

Inspired by the above papers, the aim of this paper is to establish the existence of nontrivial solutions to BVP (1.1) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear term f of BVP (1.1) is allowed to be sign-changing and unbounded from below. Secondly, the boundary conditions in BVP (1.1) are the Riemann-Stieltjes integral, which includes multi-point boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear operator. This paper employs different conditions and different methods to solve the same BVP (1.1) as [7]; meanwhile, this paper generalizes the result in [17] to boundary value problems with integral boundary conditions. What we obtain here is different from [6-20].

2 Preliminaries and lemmas

Let be a Banach space with the maximum norm for . Define and . Then P is a total cone in E, that is, . denotes the dual cone of P, namely, . Let denote the dual space of E, then by Riesz representation theorem, is given by

We assume that the following condition holds throughout this paper.

(H1) is the uniquesolution of the linear boundary value problem

Let solve the following inhomogeneous boundary value problems, respectively:

Let , , , .

(H2) , , .

Lemma 2.1 ([7])

If (H1) and (H2) hold, then BVP (1.1) is equivalent to

whereis the Green function for (1.1).

Define an operator as follows:

(2.1)

It is easy to show that is a completely continuous nonlinear operator, and if is a fixed point of A, then u is a solution of BVP (1.1) by Lemma 2.1.

For any , define a linear operator as follows:

(2.2)

It is easy to show that is a completely continuous nonlinear operator and holds. By [7], the spectral radius of K is positive. The Krein-Rutman theorem [21] asserts that there are and corresponding to the first eigenvalue of K such that

(2.3)

and

(2.4)

Here is the dual operator of K given by:

The representation of , the continuity of G and the integrability of h imply that . Let . Then , and (2.4) can be rewritten equivalently as

(2.5)

Lemma 2.2 ([7])

If (H1) holds, then there issuch thatis a subcone ofPand.

Lemma 2.3 ([22])

LetEbe a real Banach space andbe a bounded open set with. Suppose thatis a completely continuous operator. (1) If there iswithsuch thatfor alland, then. (2) Iffor alland, then. Here deg stands for the Leray-Schauder topological degree inE.

Lemma 2.4Assume that (H1), (H2) and the following assumptions are satisfied:

(C1) There exist, andsuch that (2.3), (2.4) hold andKmapsPinto.

(C2) There exists a continuous operatorsuch that

(C3) There exist a bounded continuous operatorandsuch thatfor all.

(C4) There existandsuch thatfor all.

Let, then there existssuch that

where.

Proof Choose a constant . From (C2), for , there exists such that implies

(2.6)

Now we shall show

(2.7)

provided that R is sufficiently large.

In fact, if (2.7) is not true, then there exist and satisfying

(2.8)

Since , , . Multiply (2.8) by on both sides and integrate on . Then, by (C4), (2.5), we get

(2.9)

Thus,

(2.10)

By (2.9), holds. Then (2.3), (2.6) and (2.10) imply

(2.11)

where is a constant.

(C3) shows and (C1) implies . Then (C1), (2.8) and Lemma 2.2 tell us that

The definition of yields

(2.12)

It follows from (2.6), (2.11) and (2.12) that

(2.13)

where is a constant.

Since , then (2.13) deduces that (2.7) holds provided that R is sufficiently large such that . By (2.13) and Lemma 2.3, we have

□

3 Main results

Theorem 3.1Assume that (H1), (H2) hold and the following conditions are satisfied:

(A1) There exist two nonnegative functionswithand one continuous even functionsuch thatfor all. Moreover, Bis nondecreasing onand satisfies.

(A2) is continuous.

(A3) uniformly on.

(A4) uniformly on.

Hereis the first eigenvalue of the operatorKdefined by (2.2).

Then BVP (1.1) has at least one nontrivial solution.

Proof We first show that all the conditions in Lemma 2.4 are satisfied. By Lemma 2.2, condition (C1) of Lemma 2.4 is satisfied. Obviously, is a continuous operator. By (A1), for any , there is such that when , holds. Thus, for with , holds. The fact that B is nondecreasing on yields for any , . Since is an even function, for any and , holds, which implies for . Therefore,

that is, . Take , for any , where . Obviously, holds. Therefore H satisfies condition (C2) in Lemma 2.4.

Take and for , , then it follows from (A1) that

which shows that condition (C3) in Lemma 2.4 holds.

By (A3), there exist and a sufficiently large number such that

(3.1)

Combining (3.1) with (A1), there exists such that

and so

(3.2)

Since K is a positive linear operator, from (3.2) we have

So condition (C4) in Lemma 2.4 is satisfied.

According to Lemma 2.4, we derive that there exists a sufficiently large number such that

(3.3)

From (A4) it follows that there exist and such that

Thus

(3.4)

Next we will prove that

(3.5)

If there exist and such that . Let . Then and by (3.4), . The nth iteration of this inequality shows that (), so , that is, . This yields , which is a contradictory inequality. Hence, (3.5) holds.

It follows from (3.5) and Lemma 2.3 that

(3.6)

By (3.3), (3.6) and the additivity of Leray-Schauder degree, we obtain

So A has at least one fixed point on , namely, BVP (1.1) has at least one nontrivial solution. □

Corollary 3.1Using () instead of (A1), the conclusion of Theorem 3.1 remains true.

() There exist three constants, andsuch that

Competing interests

The authors declare that no conflict of interest exists.

Authors’ contributions

All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

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